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📘 Lesson 5 of 7 · Linear Algebra

➡️ Eigenvectors

Once you know a matrix's eigenvalues, eigenvectors show you the exact directions that only get stretched or shrunk — never rotated — when the matrix acts on them.

Course progress: 71%

01 Key Concepts

Eigenvector Definition

A nonzero vector v satisfying A*v = lambda*v for some eigenvalue lambda; the matrix only scales v, never changes its direction.

Finding Eigenvectors

For each eigenvalue lambda, solve the system (A - lambda*I)*v = 0 to find the corresponding eigenvector(s).

Eigenvectors Are Not Unique

Any nonzero scalar multiple of an eigenvector is also a valid eigenvector for the same eigenvalue — eigenvectors define a direction, not a single fixed vector.

Eigenspace

The set of all eigenvectors for a given eigenvalue (plus the zero vector) forms a subspace called the eigenspace for that eigenvalue.

Diagonalization (Concept)

If a matrix has enough independent eigenvectors, it can be rewritten in a simpler diagonal form using those eigenvectors, making many calculations far easier.

02 Key Formulas

03 Solved Examples

Example 1 Find the eigenvector of [[4,1],[2,3]] for eigenvalue lambda=5.
  1. Compute A - 5I = [[4-5,1],[2,3-5]] = [[-1,1],[2,-2]].
  2. Solve (A-5I)*v=0: -x+y=0, so y=x.
  3. Choose x=1, giving y=1.
Answer: Eigenvector = (1,1), or any scalar multiple like (2,2)
Example 2 Find the eigenvector of [[4,1],[2,3]] for eigenvalue lambda=2.
  1. Compute A - 2I = [[4-2,1],[2,3-2]] = [[2,1],[2,1]].
  2. Solve (A-2I)*v=0: 2x+y=0, so y=-2x.
  3. Choose x=1, giving y=-2.
Answer: Eigenvector = (1,-2)
Example 3 Is (2,2) also a valid eigenvector for the lambda=5 case above (where (1,1) was found)?
  1. Check if (2,2) is a scalar multiple of (1,1).
  2. (2,2) = 2*(1,1), so yes, it's a scalar multiple.
Answer: Yes, any nonzero scalar multiple of a valid eigenvector is also valid

04 Practice Questions

1For [[3,0],[0,5]] with eigenvalue 3, find an eigenvector.
(1,0)
2For [[3,0],[0,5]] with eigenvalue 5, find an eigenvector.
(0,1)
3Is (3,3) a valid eigenvector if (1,1) is one?
Yes, it's a scalar multiple (3 times)
4What equation defines an eigenvector once lambda is known?
(A - lambda*I)*v = 0
5What is an 'eigenspace'?
The set of all eigenvectors for a given eigenvalue, plus the zero vector

📄 Eigenvectors — Downloadable Worksheet

10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.

🗒 Cheat Sheet ⬇ Download PDF 🖥 Interactive Worksheet